Poiseuille Flow

1.0Introduction to Poiseuille Flow

In the study of fluid mechanics, one of the most important problems is understanding how a viscous liquid moves inside a narrow cylindrical tube. This kind of flow is known as Poiseuille flow, named after the French scientist Jean Léonard Marie Poiseuille, who studied blood circulation in capillaries in the 19th century.

For JEE Physics students, Poiseuille flow is a key topic under viscosity and fluid dynamics, as it links theoretical physics with real-life applications like blood flow, oil transport in pipelines, and fluid flow in laboratory capillaries.

2.0Poiseuille Flow Definition

The Poiseuille flow definition can be written as:

Poiseuille flow is the steady, laminar flow of an incompressible, viscous fluid inside a long cylindrical pipe under a constant pressure difference.

From this definition, three important points stand out:

• The flow must be laminar (not turbulent).
• The fluid must be viscous and incompressible.
• The pipe must be long, straight, and narrow with a constant circular cross-section.

In such a flow, the fluid velocity is parabolic across the cross-section: highest at the center and gradually reducing to zero at the pipe walls because of the no-slip boundary condition.

3.0Poiseuille Flow Derivation

The Poiseuille flow derivation involves applying Newton’s law of viscosity and the balance of forces on a cylindrical fluid element inside a pipe.

Step 1: Pressure force

Consider a cylindrical tube of radius R and length L. A pressure difference ΔP is applied across the ends. The pressure force driving the fluid is:

$F_{\text{P}}=\Delta P\cdot \pi r^2$

where r is the distance from the axis of the pipe.

Step 2: Viscous resisting force

Viscous force acts tangentially due to shear stress:

$F_v=\tau \cdot A=\eta \left(\frac{dv}{dr}\right)\cdot 2\pi rL$

where η is the coefficient of viscosity, and $\frac{dv}{dr}$ is the velocity gradient.

Step 3: Balance of forces

At equilibrium:

$\Delta P\cdot \pi r^2=\eta \left(\frac{dv}{dr}\right)\cdot 2\pi rL$

Simplifying:

$\frac{dv}{dr}=-\frac{\Delta P}{2\eta L}r$

Step 4: Integration for velocity distribution

Integrating with respect to r:

$v(r)=\frac{\Delta P}{4\eta L}\left(R^2-r^2\right)$

This gives the parabolic velocity profile in Poiseuille flow. The maximum velocity occurs at the center (r=0):

$v_{\max }=\frac{\Delta P{R}^2}{4\eta L}$

Poiseuille Flow Equation (Hagen–Poiseuille Law)

“The volume rate of flow of a viscous incompressible fluid through a long cylindrical pipe under laminar conditions is directly proportional to the pressure difference and the fourth power of the pipe radius, and inversely proportional to the fluid viscosity and the pipe length.”

$Q=\frac{\pi R^4\Delta P}{8\eta L}$

To calculate the volume flow rate (discharge per unit time), integrate velocity over the cross-sectional area:

Substitute v(r):

After solving, we get the Poiseuille flow equation:

$$\begin{gathered} \text{To calculate the volume flow rate (discharge per unit time), integrate velocity over the cross-sectional area:} \\ Q={\int }_0^{R}v(r)\cdot 2\pi rdr \\ \text{Substitute v(r):} \\ Q={\int }_0^R\frac{\Delta P}{4\eta L}\left(R^2-r^2\right)\cdot 2\pi rdr \\ \text{After solving, we get the Poiseuille flow equation:} \\ Q=\frac{\pi R^4\Delta P}{8\eta L} \end{gathered}$$

Features of Poiseuille Flow Equation

• Q is directly proportional to pressure difference (ΔP).
• Q is inversely proportional to viscosity (η) and pipe length (L).
• Q increases with the fourth power of pipe radius (R4), meaning even small changes in radius significantly affect flow rate.

This is why blood circulation can be drastically affected by narrowing of arteries (arteriosclerosis).

4.0Velocity Distribution in Poiseuille Flow

The velocity distribution in Poiseuille flow is parabolic:

$v(r)=\frac{\Delta P}{4\eta L}\left(R^2-r^2\right)$

• At the center of the pipe (r=0), velocity is maximum.
• At the pipe wall (r=R), velocity becomes zero (no-slip condition).
• The average velocity is half the maximum velocity:

$v_{\text{avg}}=\frac{v_{\max }}{2}$

5.0Applications and Poiseuille Flow Example

Poiseuille flow has several practical and exam-related applications.

• Blood Circulation: Blood flow in veins and arteries follows Poiseuille flow principles. A slight narrowing of arteries greatly reduces blood flow due to the R4 dependence.
• Oil and Gas Pipelines: Engineers use Poiseuille’s law to estimate energy loss and design efficient pipeline systems.
• Capillary Tubes in Labs: Used to measure viscosity of liquids using Poiseuille’s equation.

Poiseuille Flow Example (JEE Style Problem)

Problem:
A liquid of viscosity flows through a capillary tube of radius 0.5 mm and length 0.2 m. The pressure difference across the ends of the tube is 1 Pa. Calculate the volume of liquid flowing per second.

Solution:
Using Poiseuille flow equation:

$$\begin{gathered} Q=\frac{\pi R^4\Delta P}{8\eta L} \\ =\frac{3.14\times (0.5\times 10^{-3}{)}^4\times 1}{8\times 2\times 10^{-3}\times 0.2} \\ Q\approx 1.23\times 10^{-11}{\text{m}}^3/\text{s} \end{gathered}$$

Thus, the liquid flow rate is extremely small, showing the strong effect of tube radius on flow.

6.0Key Electromagnetic Field Properties

Important electromagnetic field properties include:

• Dual nature: Electric and magnetic fields are interdependent; a changing electric field induces a magnetic field, and vice versa.
• Superposition: Multiple fields from different sources add vectorially without interference.
• Transverse wave behavior: In electromagnetic waves, the electric and magnetic fields oscillate perpendicular to each other and to the direction of wave propagation.
• Relativistic nature: Observers in different frames may perceive a combination of electric and magnetic components depending on their motion.